Yevhen Konoplyanka scores for Ukraine in the 1-1 draw at Wembley a year ago – a similar scoreline on Tuesday would favour England. Photograph: Tom Jenkins

Out of chaos has come hope. When Oleh Blokhin quit as national coach of Ukraine after the 1-1 draw against England at Wembley to take charge of Dynamo Kyiv, there were three schools of thought. One thought it an outrage that anybody should abandon what was perceived as a patriotic duty; one accepted that, having spent 17 years as a player at Dynamo, the emotional pull was too strong to resist; the other breathed a sigh of relief at the deposition of a cranky and authoritarian manager whose teams had played crabby, bad-tempered football. As Dynamo's slow decline continues – they lie sixth in the table after eight games and, as Metalist Kharkiv and Dnipro Dnipropetrovsk grow, are arguably not even Ukraine's second side (after Shakhtar Donetsk) any longer – it is the third school that seems most reasonable.

Andriy Bal, the defender whose nightmare effectively cost the USSR their last-16 tie against Belgium in the 1986 World Cup, took over as caretaker, but he oversaw a draw in Moldova and a home defeat to Montenegro.

At that point, Ukraine's race looked run. But Mykhaylo Fomenko – yet another who played under Valeriy Lobanovskyi at Dynamo and for the national team – took over in December and issued his equivalent of Imran Khan's "cornered tigers" rallying cry to Pakistan at the 1992 cricket World Cup. He urged his team to play each game as though it was their last, and the response was immediate: two goals in the first seven minutes in Warsaw set Ukraine on their way to a 3-1 win over Poland and reignited their campaign. A 9-0 victory over San Marino on Friday made it four wins out of four in competitive games.

"Having taken charge of the team, he had many conversations with the players," said the goalkeeper Andriy Pyatov. "In general, he loves it. He often asks how you are doing, he tries to understand our mentality. You can already see that he's found a common language with us. He has his own approach regarding organisation of the game. He demands discipline on the pitch, commitment when challenging for the ball. The fact that we are accomplishing his demands is reflected in our results in the last few games. So, the coach is doing everything right."

There is clearly not too much that can be read into a win over a side as weak as San Marino, but there was something impressively relentless about the way Ukraine dismantled them, at the way they kept chasing goals to reduce England's advantage in terms of goal difference to seven, even the fact that eight different players found their way on to the scoresheet.

This is a team with a varied approach, more fluid and adaptable than some Ukrainian teams of the past. England will have noted the quality of Yevhen Konoplyanka's corners, which led to four of the nine goals, and also Yaroslav Rakitskiy's aptitude from free-kicks.

Rather more encouraging from England's point of view is that Ukraine's goalkeeper remains Pyatov, who is not the most commanding under the high-ball and is prone to occasional nightmare games, as Chelsea found to their advantage against Shakhtar in the Champions League last season.

The victory over San Marino was the biggest in Ukrainian history, but Fomenko seemed unbothered by that. "We should forget this game as soon as possible and think about England," he said. "It's good that no one has been injured. We are already living the game against England and that match [against San Marino] helped us prepare well. They beat Moldova with ease and that means something."

That may be so but it does not alter the fact that for both teams Tuesday night is crucial. Win and England are almost in Rio; lose and they are probably facing the play-offs at best. And the momentum is very much with Ukraine.